Effective mass of an electron

We have demonstrated that the wave function v / is a linear combination of N periodic functions, and we liave seen that we would be well-advised to use a distribution of the wave vector s values. If N tends to infinity and v / is expressed  

We now have a wave packet whose maximum is characterized by the multiple waves being in phase, which means that the quantity y = 27rvt — kx does not 

The term xlt has the dimension of velocity it represents the velocity V of the wave packet and therefore, the speed of the particle. 

On the other hand, we know that E = hv. It follows that  

In the presence of an external force F taking its origin from, for instance, an electric field, we will write  

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The energy E will necessarily have this minimum, but its value at this point can be positive or negative only in the latter case will a stable self-trapped particle (i.e. a small polaron) form. This is most likely to occur for large effective mass, and thus for holes in a narrow valence band or for carriers in d-bands. If the polaron is unstable then there is practically no change in the effective mass of an electron or hole in equilibrium in the conduction or valence band. 

The idea of the effective mass of an electron is known not to be strict in particular, the definition of the effective mass depends, generally speaking, on the physical phenomenon under consideration. For instance, when one analyses electron motion in a periodic crystal, the effective mass is usually determined as [13]... 

According to Einstein s special theory of relativity, the effective mass of an electron increases whenever its speed becomes a significant fraction of the speed of light, c 3 X 10 m/sec. Specifically,... 

More direct determination of electron effective masses was first performed by Kaplan et al [16] using electron cyclotron resonance (ECR) in n-type 3C-SiC epitaxially grown onto a silicon substrate. They obtained transverse effective mass m t = (0.247 0.011) n, and longitudinal effective mass m, = (0.667 0.015) m0. The effective masses derived from cyclotron resonance agree, within experimental error, with the values obtained from Zeeman luminescence studies [7] of small bulk crystals. An average effective mass of an electron, given by the equation me = (m t2m, )l/3, is 0.344m0. Recently, similar ECR measurements were made by Kono et al [17]. 

As the effective mass of an electron in silicon is approximately one tenth of the electron rest mass, and as the relative permittivity of silicon is about 10, the energy needed to free the electron is about one hundredth of the band gap, which suggests that the electron should be very easily liberated. Donor energies are often represented by an energy level, the donor level, drawn under the conduction band (Figure 13.9a.ii). 

Calculate the donor energy-level position in silicon doped with phosphorus using the Bohr model . The effective mass of an electron is 0.33me and the relative permittivity of silicon is 11.7. 

Notice that the effective mass varies as the curvature of the band. In the conduction band in the vicinity of k = 0 where the band is curved upward, the effective mass of an electron is positive. However, near an inflection point the effective mass becomes infinite, and then becomes negative in the regions where the band curves downward. It follows that electrons in the second or valence band near k = 0, have negative mass. 

Quantum confinement is manifested when carriers in a semiconductor are confined by potential barriers to regions of space less than or equal to the de Broglie wavelength or the Bohr radius of excitons in the bulk material.Quantum dots (QDs) are confined in all three dimensions. For example, bulk CdSe has a band gap of 1.7 eV, the effective mass of an electron is 0.44 mo, the effective mass of a hole is 0.13 mo and the Bohr radius of the CdSe exciton is 5.6 nm. Particles... 

In other words, the effective mass of an electron Bloch wave is inversely related to the curvature of the E(k) diagram. Strong curvatures mean small effective masses and vice versa. The effective mass is generally expressed in units of electron rest mass in a vacuum, i.e. m /mo. 

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